About Rationally Speaking

Rationally Speaking is a blog maintained by Prof. Massimo Pigliucci, a philosopher at the City University of New York. The blog reflects the Enlightenment figure Marquis de Condorcet's idea of what a public intellectual (yes, we know, that's such a bad word) ought to be: someone who devotes himself to "the tracking down of prejudices in the hiding places where priests, the schools, the government, and all long-established institutions had gathered and protected them." You're welcome. Please notice that the contents of this blog can be reprinted under the standard Creative Commons license.

Thursday, November 01, 2012

Can a statement be simultaneously true and false? That might seem like sheer nonsense to you -- but not to certain modern logicians.

In this episode Massimo and Julia are joined again by philosopher and logician Graham Priest, who explains why we have to radically revise our notions of "true" and "false." In the process, he explains classic puzzlers like the "barber paradox": "In a village, the barber shaves all men who do not shave themselves. Does he shave himself?"

Follow along for an episode that really takes to heart the podcast's tagline: exploring the borderlands between reason and nonsense.

No, it's a paradox and was known to be so before Goedel's theorem. Undecidable statements (like the continuum hypothesis under the ZFC axioms of set theory) aren't paradoxical, they just can't be proven true or false.

And jose, if he lets his beard grow, then he doesn't shave himself. Which means he shaves himself.

Some undecidable statements, like Liar sentences, certainly are paradoxical (in this case, true iff false). The continuum hypothesis isn't because it's merely independent of ZFC (i.e., ZFC is consistent with both the continuum hypothesis and its negation).

It's a grammatical error. The relation of barber to all men is as their barber, so he is excluded from the group 'all men' by logical application of the grammar to the real relation. Logic rules when grammar fails.

This is about logical error built into the use of grammar. If you can't see that, too bad. There are many ways illogical propositions can be presented, including grammatically.

For other readers, the solution is to interpret propositions logically. The only reason the barber example seems 'paradoxical' is because the reader ASSUMES the barber is included in 'all men', which he cannot be, so modify the statement to a logical assumption.

Piggybacking on what Steve said. The Barber Paradox is a loose version of a more general paradox, Russell's Paradox, which is itself related to a paradox identified by Georg Cantor in 1899. Cantor's paradox is, crudely, that there are sets than which none could be larger, and given any set there is one larger than it. Russell related a revision on this theme to the work Frege on the foundations of mathematics: Most sets are not members of themselves. E.g., the set of all Irishmen is not itself an Irishman, and so is not a member of the set of all Irishmen. But consider sets that are not members of itself and ask if it is a member of itself and we immediately encounter a contradiction. Let S be the set of all sets that are not members of itself. If S is a member of S, it is a member of the set of all sets not members of themselves, so S is not a member of S. But if S is not a member of S, it is not a member of all sets not members of themselves, and thus it is a member of itself: S is a member of S.

Here's how it works. There is self-consistency. It is fundamental logic. If you don't have it, you have self-contradiction. Easy.

You can propose anything as lacking self-consistency, such as 'I always tell lies and I always tell the truth', Illogical nonsense. You can shorten that to 'I always tell lies' by making the statement itself a part of the proposition rather than objectifying it. Easy.

No 'paradox'. unless a paradox is basic self-contradiction expressed objectively or subjectively. Easy to do and analyze as nonsense, without invoking set theory as more abstract nonsense. Nothing at all, in fact.

your recent comments to the effect - if I understand your convoluted phrasing correctly - that every logician who talks about paradoxes has just been fooled by a grammatical mistake shows at once your profound ignorance and your just as large degree of unjustified self-confidence. Mind blowing, really.

Has anyone studied the extent that paraconsistent logic, (as opposed to logics featuring explosion) is natural and intuitive for humans? Despite the seemingly widespread intuition that a contradiction just *cant* be both true and false, I wonder if people are more tolerant of contradictions than we realize at first blush.

My family is religious in the mainline protestant, wishy washy way of many religious liberals, and at the same time my brother and mother are both doctors and quite scientifically literate. For me, personally, we already know enough science to rule out a personal, theistic god, at least of the sort that has anything to do with Christianity. My brother, at least, admits that he can see the logic of this, but says that he prefers to 'keep some mystery around' and doesn't feel comfortable extending scientific ways of thinking throughout his belief structures.

So is mystery functioning as one of the 'logical singularities' Graham mentioned, allowing religious beliefs to be dependent on only a small set of nonthreatening 'relevant' beliefs, with the dangerous epistemic values of science cordoned off as 'irrelevant'? Has anyone else encountered this kind of thinking in their religious friends? Are atheists (and possibly fundamentalists) naturally inclined to be more hostile to paraconsitencies, naturally inclined to make *all* of their beliefs consistent?

Has anyone studied the extent that paraconsistent logic, (as opposed to logics featuring explosion) is natural and intuitive for humans? Despite the seemingly widespread intuition that a contradiction just *cant* be both true and false, I wonder if people are more tolerant of contradictions than we realize at first blush.

My family is religious in the mainline protestant, wishy washy way of many religious liberals, and at the same time my brother and mother are both doctors and quite scientifically literate. For me, personally, we already know enough science to rule out a personal, theistic god, at least of the sort that has anything to do with Christianity. My brother, at least, admits that he can see the logic of this, but says that he prefers to 'keep some mystery around' and doesn't feel comfortable extending scientific ways of thinking throughout his belief structures.

So is mystery functioning as one of the 'logical singularities' Graham mentioned, allowing religious beliefs to be dependent on only a small set of nonthreatening 'relevant' beliefs, with the dangerous epistemic values of science cordoned off as 'irrelevant'? Has anyone else encountered this kind of thinking in their religious friends? Are atheists (and possibly fundamentalists) naturally inclined to be more hostile to paraconsitencies, naturally inclined to make *all* of their beliefs consistent?

This is fascinating stuff. It's interesting that there really is a plurality of logics. In this case, by weakening one or more rules of inference so that you can no longer prove any arbitrary statement from contradiction, and so you eliminate explosion. The result should be a weaker logic that renders many other things unprovable from basic axioms. On the other hand you have set theorists such as Hugh Woodin exploring stronger logics (in particular, Omega logic) in an effort to decide questions such as the Continuum Hypothesis. So currently there are motivations in two opposite directions.

I suspect that there are no sentences that are both true and false, as long as we restrict ourselves to sentences that are grounded in material reality or that are grounded in some set of axioms. For ungrounded, self-referential sentences, I'm not convinced that dialetheism is necessary, but it seems to provide a tenable resolution to the liar paradox, Russell's paradox, and so on. It just isn't the only one.

More subtle (and non-paradoxical) sentences lead to Godel's incompleteness and Tarski's undefinability theorems. I don't see paraconsistency as having any particular bearing in this area. The Godel sentence isn't "true and false", nor is it "neither true nor false". It is simply true.

On the other hand I still believe that there are plenty of statements that are neither true nor false, the Continuum Hypothesis being a prime example.

Propositional logic can be shown to be consistent and complete using its own rules, but, as Gödel showed, any higher order logic that includes the natural numbers cannot be shown to be consistent from within itself, and if the higher order logic is assumed to be consistent, then it can be shown to be incomplete. Were that not the case, then the continuum hypothesis or the axiom of choice, its equivalent, being well-defined statements in Zermelo-Fraenkel set theory, could be proved or disproved by the rules of ZF. Nevertheless, it is not strictly correct to say that the continuum hypothesis is neither true nor false, but rather that it is undecidable from the other azioms of ZF; one can choose either its affirmation or negation as axiomatic and maintain the consistency of ZF in either extension. Mathematics would be the poorer were it to negate the continuum hypothesis, but it would not be inconsistent unless it were so already. Similarly, one can have parallel lines meet in a non-Euclidean geometry that is as consistent as the Euclidean variety where they never meet. That is the nature of independent axioms. They can be assumed to be either true or false, though never both in the same mathematical system.

While ZF set theory is perforce incomplete and not provably consistent, the earlier set theory, created by Cantor, is demonstrably inconsistent, as shown by Bertrand Russell more than a century ago in what is known as Russell’s Paradox, which is similar to the liar paradox in that it arises from a statement reflexively referring to its own negation. Gödel’s proof of incompleteness of higher order logic, using a version of Cantor’s diagonal argument, also refers reflexively to a negation of a property of itself, but avoids paradox by affirming a statement’s own unprovability rather than its falsity, and Cantor creates a mathematical object by assuming it does not exist.

Though ZF rescued mathematics from demonstrable absurdity, it did so by adding iterative rules by which sets can be legitimately defined, similar to Whitehead and Russell’s earlier, more controversial, attempt in Principia Mathematica. The legitimacy of the ZF rules can be, and are, debated by philosophers of mathematics thanks to the inherent restrictions placed by Gödel on what one can say about mathematical-logical consistency, though mathematicians have done fine with ZF and its extensions.

Early mathematical logicians to take issue with the law of the excluded middle were the Intuitionists, headed by Brouwer, who proposed a constructivist approach to certain mathematical conjectures involving infinite sets. According to the Intuitionists, if one defines a mathematical object in such a way that one cannot deductively discover whether or not it exists, then it is illegitimate to say that it either exists or does not.

Gödelian incompleteness, unprovability of consistency, undecidable propositions such as the axiom of choice, and Intuitionist-constructivist rejection of the excluded middle are all very different from the dialetheist assertion that some statements are both true and false. A limited form of set theory can even live with the Intuitionist version of no excluded middle, but it dies a quick death with the dialetheist version. I do not see an obvious reason why ordinary language cannot live with dialetheism, but I also cannot see a convincing reason why it should.

Actually, my last statement about the CH was just a statement of belief, not a statement of certainty. It is true that statements can be strictly true (or strictly false) and still be undecidable. But CH is not strictly true nor strictly false in either ZF or ZFC (I'm a bit confused by your statement labeling AC the equivalent of CH), because we know there are models of ZF satisfying each case. That is also true of AC. ZF isn't strong enough to pin down a truth value. However, I think set theorists are more interested in whether CH or AC holds in the von Neumann universe V, and it is this question I was referring to. Does adopting AC or its negation eliminate V as a possible universe of sets, or do we merely get a different V under ZFC than we get under ZF(not C)? And how do either of these differ from the V we would get under ZF? I don't know the answer but I suspect that they must be different. At the very least, they would have to disagree over whether some function constitutes a choice function, or whether some set is in the domain of that function. And if that disagreement exists, it may be that AC is more than merely undecidable. Likewise for CH.

I think there is a difference between the AC and CH questions and (familiar) questions about the natural numbers. Suppose it turns out that the twin prime conjecture is undecidable. I do not think that would mean that TPC is "neither true nor false". In order for TPC to be neither, we would need to have the possibility of two models of arithmetic disagreeing over whether some numbers are prime.

In short, I don't think undecidability is a sufficient condition for "neither true nor false", but it is a necessary one. I don't think we can discount the possibility of "neither true nor false" statements existing, and while that view may be popular with intuitionists, it does not (as far as I'm aware) require an intuitionist view to admit this.

By AC and CH being equivalent I mean that generalized CH can be derived from AC and not GCH can be derived from not AC. You may be right to say that AC has no well-defined truth value in ZF, but AC is by definition true in ZFC and V. Since the law of excluded middle can be derived from AC, it would seem that in ZFC and V there cannot be statements that are neither true nor false, only, perhaps, undecidable propositions where the truth value cannot be determined but is one or the other, and, by necessity, Gödelian true but unprovable monsters.

I don't think you can derive GCH from AC, nor not GCH from not AC. For the former, just consider that CH is already independent from ZFC. Therefore GCH, which implies CH, cannot be proven from ZFC (if ZFC is consistent). I think GCH does imply AC, but GCH is stronger than CH so we do not have CH implies AC. At any rate, without the other half, you do not have equivalence.

I'm not familiar with the "AC implies LEM" proposition, but a quick search turned up some references confirming this. One source indicated some additional assumptions were required. Wikipedia stated that AC is sufficient. Not having time to study this at the moment, I don't have an opinion as to which source is correct. If the Wikipedia article is correct then it certainly changes the view -- when we assume AC. But AC is a powerful axiom, and you can prove a lot by assuming it...such as the Banach-Tarski paradox. AC is a very productive axiom, but I tend to see it as another "neither true nor false" statement. What this means is that we can adopt AC as an axiom and do meaningful math, and we can adopt not AC as an axiom and do meaningful (but different) math.

Correction...If the Wikipedia article is right and AC is sufficient to prove LEM, then AC must be one of "true or false". Because if it is neither true nor false, then LEM doesn't hold, so AC must be false -- contradicting the premise. However, if additional assumptions are required besides AC, then it is still possible for AC to be neither. But I don't see either of these scenarios as proving that CH is subject to the law of the excluded middle.

Sorry, I think my correction was too hasty. If AC proves LEM, then the proof is only valid if AC is true. That is, LEM must hold if AC is true, and may or may not hold if AC is not true. AC is not true in two cases: (1) AC is false, and (2) AC is neither true nor false. So, "not LEM" is compatible with AC being neither true nor false.

Yes, one may reject LEM, as did the intuitionists, ending up with a limited mathematics. It was a price the intuitionists were willing to pay on meta-mathematical grounds, by claiming that even in a mathematical system assumed to be consistent such as ZF, not every infinite set the mind constructs can be said to exist or not exist. This rejects AC, which explains why intuitionists have remained a negligible minority among mathematicians. The acceptance of AC, and hence LEM, by most mathematicians implies a meta-mathematical affirmation of the legitimacy of asserting existence (or non-existence) of well-defined mathematical objects.

Our discussion is off-topic and better belongs, perhaps, on an earlier thread on Platonism. But I can say nothing about dialetheism in the context of set theory because it speaks about language, not mathematics. I do not think contradictions are true even in ordinary language, and if dialetheists are happy with the idea, it’s their problem.

Thanks for the reference to reality, I have been trying for a week or more to steer discussions that way. The reality of this conflation of abstract paradoxes into prominence is the aplication of logic to contradicting theories. Theories that overlap and contradict in some way to falisfy one or other or both. Propositions that are not logically consistent. Neither is true or false until reconciled, as as theories they are always open to challenge about truth anyway. It's called the scientific method, not paraconsistent conflation.

I think you were replying to my post, so thanks for (I think) agreeing with me, but you might have missed a point or two.

I'm not sure what you think is being conflated with what. I don't see any conflation.

The purpose of logic is to partition statements of some language (that of mathematics, physics, chemistry, biology, natural language, or whatever) according to truth value, given some starting assumptions. Along the way, it must be able to resolve paradoxes, abstract or not, even if those paradoxes are side effects of the grammar. That can be done axiomatically (using the Axiom of Foundation to restrict the domain, for instance) or by restricting inference (paraconsistency).

Paraconsistent logic is weaker than classical logic, so its power of proof is reduced. That doesn't make it unsound. If I understand it correctly (and it's new to me, so it's possible that I don't), paraconsistent logic doesn't *introduce* contradictions; it can't, because it can only prove a subset of what classical logic can prove (given the same set of axioms in either case). But paraconsistent logic does allow adoption of *stronger* axioms (if we are talking about abstract theories) or empirical premises (if we are talking about material theories) that might lead to inconsistency (explosion) under classical logic. Is it true or false that Schrodinger's cat is alive (the instant before we look in the box)? Maybe it can be both, provided that the rules of inference don't lead us into trouble as a result.

In short, you can have strong logic + weak axioms, or you can have weak logic + strong axioms, but you can't have strong both. Dialetheism is just opting for the latter approach instead of the former. I'm not ready to jump on that train right away, but I can't find anything inherently unsound about it, and I like the idea that such an alternative exists.

I would just take the realistic option you present above, basic logic, rather than attempting to raise uncertainty or self-contradiction to status as any true and false reality. that's not reality, its just an abstract game. Use logic in pursuit of the scientific method. Ignore the offensive non-replies to my post by others below, they have no arguments to make in their replies - they are purely offensive - so continue to argue objectively and don't get sidetracked by that.

Honestly, Dave, I have a hard time parsing what you are saying. I completely understand Massimo's and Eamon's frustration. You talk about conflation but never make it clear what things are being conflated. And now "attempting to raise uncertainty of self-contradiction to status as any true and false reality"... I have no idea what this means. A little attention to sentence structure would go a long way here.

I don't care if you understand their frustrations, I have a clear argument based entirely on logical self-consistency which none of you have answered by various illogical arrangements of terms as staements. There is no excuse for personal attacks, keep the attacks to the argument, call the argument ridiculous if you like, but don't call me preening, egotistical etc. as they do, that would be a mistake.

So as I have some time, let me explain how to interpret my clear statement. Priest argues that there is a true and false status to paradoxes, when in fact every single paradox is based on self-contradiction, its the deifiniton of a paradox. If a real theory has contradicting elements, or two theories contradict, it doesn't mean they are both true, it means the truth is uncertain until falsification. I write in brief, but surely you read my brief statement as having that meaning in the context of prior posts and discussion in general. It should not require me to lay it out chapter and verse like this, between reasonable people.

I'm afraid I have to second Eamon here. DaveK, your latest comments on this thread have been embarrassing, they only show the depth of your ignorance, which is apparently matched by an equally large ego.

Not a single issue addressed in reply. This is a tendancy, leaving only offensive comment. It also arose in the thread on emergence. You should regulate manners here, avoid ad hominems, and stay on track with the issues, as found on other public blogs.

I'm saying the barber cannot, logically, be one of the group of 'all men'. Believe it or not, sensible interpretations of such statements are made in courts all the time to preserve some value to the evidence.

this isn't a question of ad hominem. It's that it is impossible to answer your argument because you don't have an argument. You just don't seem to get it, which would be fine if you at least had the modesty and good sense of admit it. Which you don't.

Of course its ad hominem to say someone is ignorant, pompus, preening, egotistical etc (read the above). Don't try to redfine reality as you do with useless conundrums about emergence and paradoxes. Deal with what I say. If I have said something ad hominem or directly offensive and you and Eamon have done, point it out specifically. This is a dsigrace as a public blog. Establish any of the ad hominem claims if you can: you cannot beacuse they are filthy lies.

Re 'I'm saying the barber cannot, logically, be one of the group of "all men".'

Before continuing with my comment, I should like to clarify that 'group', 'class', & 'set' are not, strictly speaking, interchangeable. But, unless you mean something other than 'set', I will interpret 'group' to mean 'set'.

That said, in logic we define 'set' as (crudely) a well-defined collection of distinct objects – that is, a collection of objects that satisfy some condition for set-inclusion. [Our definition of fuzzy sets would differ from the preceding.] We can define sets in any number of ways & we can populate different sets with the same objects. [E.g., you are a member of the set of all h. sapiens and a member of the set of all Rationally Speaking blog commentators.] Hence, we can identify the set of all regular contributors to the Rationally Speaking blog, the set of all Irish-born American logicians, the set of all human beings who live on a certain island, etc.

Now take a town wherein all the men are cleanly shaven. They either shave themselves or they do not. If they do not shave themselves, the barber does it. Thus, from this we can define two sets: the set of all men who shave themselves and the set of all men who have the barber do it. Now take a male barber. He is clean-shaven. Thus, he either shaves himself or he does not. If he does not, he shaves himself – since the barber shaves all and only those men who do not shave themselves. But since he does shave himself, he doesn't – since the barber shaves all and only those men who do not shave themselves. Thus, he shaves himself and he does not shave himself. QED.

Now you want to say there is something wrong with this picture. What, exactly, is it? We defined the sets in an appropriate way and our reasoning is valid. Certainly you cannot simply deny that there is a problem because it is unseemly (and it IS unseemly). So, please demonstrate where I have gone wrong.

Maybe (and just maybe) the point DaveK is trying to rather brusquely get across is that the barber paradox isn't exactly paradigmatic, as its deviance depends on how one logically interprets the sentence. Consider the following two interpretations:

B = is a barber S = shaves

(1) ∃x[Bx ∧ ∀y(¬(ySy) ↔ (xSy))]

(2) ∀x(¬(xSx) ↔ ∃y(By ∧ ySx))

(1) says, "There is a barber such that he shaves any man if and only if they don't shave themselves." (2) says, "Any man doesn't shave himself if and only if there is a barber such that he shaves them." Clearly only (1) could ever be paradoxical, namely, when 'x' and 'y' are substituted for the same object (when the barber shaves himself). (2), however, is perfectly fine. Of course, one could argue that (2) is the logical form of a completely different sentence altogether. Nonetheless, it's difficult to deny that (2) is very close in meaning to (1), and has the added benefit of being paradox-free.

Anyway, the above is my charitable interpretation of what DaveK mean by "using logic in pursuit of the scientific method."

Well, usually when a statement leads to inconsistency, we conclude that the statement is false. Thus (1) is simply false. I don't think a counterfactual statement is a bona fide paradox when there is nothing to compel us to accept it as true.

I was thinking the barber "paradox" was an attempt to exemplify Russell's paradox in concrete terms. And just as the resolution of the barber paradox is simply to deny the existence of the barber, the resolution (one possible resolution, at any rate) of Russell's paradox is to deny the existence of a "set of all sets that do not contain themselves". But there is nothing in naive set theory to prevent this set from existing, so Russell's paradox is a legitimate paradox in highlighting a problem of naive set theory. And so we have the Axiom of Foundation, which prevents any set from containing itself. This means the "set of all sets that do not contain themselves" would be the same as the "set of all sets", but it also means that the "set of all sets" cannot exist either (such a collection must be a proper class). Perhaps the weaker logic of paraconsistency can eliminate the need for the Axiom of Foundation (which is a limiting axiom), but I don't know enough about it to say if this is so.

In the barber case, we don't need the Axiom of Foundation to avoid a paradox, because we can search the Universe over and never find such a barber. Likewise, I don't think we need to resort to dialetheism to fix the barber "paradox".

On the other hand, the liar paradox is a problem of a different sort. If we define "L(x)" to mean "x is a lie", then the paradox amounts to "y = L(y)", which can be expanded to "y = L(L(L(... ...)))". Notice that, if we take this to represent the "complete" (omega-length) expansion of y, we can't actually write the symbol "y" anywhere on the right-hand side of the equation. It has to appear to the right of an infinite number of "L(" tokens, and to the left of an infinite number of ")". y is not a well-founded sentence, so we can't declare it either only true or only false. At best, it is neither, or (in the dialetheist view) both. Another way to look at it is that y is not in the domain of the predicate L, so L(y) has no meaning.

Your statement (2) is not even close to the barber paradox. It is saying "Every man who does not shave himself has a barber who does it for him". If a barber does not shave himself, there exists a barber somewhere who shaves him. Interestingly, no barber can shave himself if this sentence is true (if "xSx" is true, then "not xSx" is false, and the double arrow requires "there exists a barber who shaves x" to be false as well, but if xSx is true and x is also a barber, then there does exist a barber who shaves x).

∀x(xSx) ↔ Bx says, "Every man shaves himself if and only if he is a barber." This completely avoids the paradox, as everyone in the village just ends up shaving themselves (and being barbers) or not shaving themselves (and not being barbers).

Citing empirical matters in order to falsify (1) is insufficient. The barber paradox may not be the most robust of logical paradoxes, yet it is still a logical paradox, and therefore demands a logical solution. The conclusion that the barber never shaves himself is acceptable in this context given the domain of discourse. Though I should have made these two points explicit, the quantifiers in (1) and (2) only range over the 'village' people, and the existential quantifier is a definite description (i.e., it singles out a unique person in the village, the barber). (2) resolves the problem in (1) by logically forcing us to accept that the village barber can't shave himself. That's not to say that our barber can't go to some other village to shave himself or get shaven; he just can't do it at home. Moreover, (2) isn't too far in meaning from (1), as it still implies, "In a village, the barber shaves every man who doesn't shave himself." But Regarding the follow-up, "does the barber shave himself," (2) doesn't face the insuperable problems (1) does.

"Citing empirical matters in order to falsify (1) is insufficient." So if I were to claim that there exists an even prime greater than two, logic is forced to reconcile the resulting contradiction?

And no, (1) and (2) really aren't that close. The quantifier shift is more profound than you are admitting: (2) allows more than one barber to do the shaving of men who don't shave themselves. And this means the premise does not require that there is a single barber who shaves all the men who don't shave themselves. You can have the following partition of the village as a result of this relaxation: non-barbers who shave themselves, non-barbers who are shaved by barbers, and barbers who shave each other.

If you don't accept the solution that (1) is simply counterfactual, and require that "it is still a logical paradox, and therefore demands a logical solution", and if you are suggesting that changing the problem statement from (1) to (2) is that solution, then you can also win the Kobayashi Maru scenario.

Suppose there is a man, x, who shaves himself. Then the second part implies Bx. Therefore we can satisfy (By ∧ ySx) by letting y = x. Since y exists, we must have ¬(xSx), contradicting our supposition. The only way to avoid this contradiction is to negate the supposition:

¬∃x(xSx)

But wait. That is equivalent to

∀x(¬xSx)

and by reapplying the first premise, we must conclude that

∀x(∃y(By ∧ ySx))

which says that at least one barber exists. By the second premise, that barber must shave himself, which we have just shown to be impossible.

In English, the first statement is "Every man who doesn't shave himself has a barber who does it for him, and every man who has a barber to shave him does not shave himself". Since the arrow goes both ways, anyone who is shaved by a barber does not shave himself. So a barber cannot shave himself, because if he does, he is shaved by a barber. If a barber cannot shave himself, this contradicts the second statement, "Every man who shaves himself is a barber and every barber shaves himself". So barbers can't exist in this scenario, and all men must shave themselves. But only barbers shave themselves!

In this case, the only way out is to conclude that the domain that x is chosen from is the empty set. In other words, there are no men in the village at all.

I didn't mean to write unclear sentences, just obvious ones. I was only saying that self-consistency rules both in logic and in reality. Whether a statement is grammatically or mathematically logical, it must be self-consistent or it remains inconclusive. In the Scientific Method, theories are inconclusive if they are inconsistent in a point of overlap, and that's how any inference should be viewed if it offends self-consistency. It is not both true and false, it is conditionally either/neither until the illogicality and unreality is resolved.

I realize you have a technical approach to the general problem of paradoxes, with Russell and other references, but it boils down to self-consistency. Perhaps you are trying to penetrate Cantor to determine whether set theory is or is not logically self-consistent. That just returns to the task of penetrating any 'statement' presented on given terms to be assessed by logic applied to those terms. The more complicated the terms (or the paradigm in general if it goes that far), the more difficult the assessment of the illogicality. Endless fun for the young at heart.

@Eamon,"Now take a town wherein all the men are cleanly shaven. They either shave themselves or they do not. If they do not shave themselves, the barber does it."Forgetting that they, including the barber and any trained apes or robots available, in reality, if they shave at all, may also shave each other.

I would tend to agree with those that believe dialetheism is a bit much. I know the podcast mentions that we may have a very intuitive notion of the law on noncontradiction that might not be correct, but the onus is on those who are dialetheists to show what exactly it means for something to be both true and false at the same time. Julia's resistance to the idea, at least in my opinion, is very warranted. I do not want to come off seeming like someone who is not open; I do believe that paraconsistent logics could be useful for things such as information processing and programming, where errors that lead to explosion should be avoided. However, and this is also touched upon in the podcast, that in no way shows that paraconsistent logics are somehow the ultimate logic because of that usefulness. I think there are many other avenues that can be used to answer the liar paradox. Massimo, I know you have mentioned that you are a platonist in regards to mathematics, which I think is a very defensible position and which I myself subscribe to. How would you combine the idea of pure mathematical truths with statements that are both true and false at the same time? I do not mean to suggest that you believe dialetheism to be true, but you do seem to entertain the notion if I am not mistaken. I myself am nowhere near believing that position, but I would like to hear your thoughts.

I'm not sure how to reconcile dialethism with mathematical Platonism, but I'm also on the fence about accepting the first one, and at any rate math isn't exactly the same thing as logic.

Baron et al.,

to say that "this wouldn't happen in reality" is to completely miss the point. It is a logical paradox, and logical reality is much broader than physical reality. It is no objection at all (indeed, as Richard put it commenting on Dave, it is *irrelevant*) to point out that no real town would have a barber who behaved like that.

Dave,

> I'm saying the barber cannot, logically, be one of the group of 'all men'. <

I didn't say it is irrelevant to claim no such village exists; that might have been someone else. I can't agree with DaveK (honestly I can't, because I have no idea what he is saying!) but I later made the point that the barber paradox, as stated, is simply a counterfactual statement. Most logicians (I believe intuitionists are an exception) take contradiction as proof that the premise is false. Reality does not permit "a barber who is a man, and who shaves those men in his village, and only those men, who do not shave themselves". Yet this doesn't work for the Russell paradox, a statement with essentially the same form, if we are working in naive set theory, because naive set theory has no restrictions on which collections are sets. If we add a restriction (Axiom of Foundation) then we get a set theory that models a reality in which the paradoxical barber cannot exist, and the statement of the Russell paradox becomes a simple counterfactual statement.

Logical paradoxes are 'real' problems because logic is part of the 'real' world (I imagine this will lead to further confusion, but who cares, eh?).

In any case, Massimo is right that you are missing the point. In point of logical fact, the Barber Paradox is offered only in order to explicate a more general problem first identified by Georg Cantor in his work on set theory.

Eamon, The barber paradox was at the center of the podcast, so your attempt to insert your own reasons for mentioning it at this late date is disingenuous to say the least. Nobody was questioning the point for those assertions, were they?I stated to Massimo that, "Eamon thought he was presenting a real problem even if you didn't."And you've confirmed that.

You two can continue to divert the issues until the cows come home, but it doesn't change the fact that all paradoxes make the pretense of being real problems.

Three things to say here. First, in my initial comment on this thread, made on 11.1.2012, I mentioned that the Barber Paradox is related to a more general paradox identified by Georg Cantor. The implication is of course that the Barber Paradox serves simply as an explication.

Second, you again miss the point. In the podcast, Priest mentions certain paradoxes which might be good candidates for dialethias (true contradictions). These paradoxes stem from self-reference and the types of antinomies identified by Cantor and others. The problem is not that there is actually IS an island or town wherein all the men are cleanly-shaven, etc., and that there is one man, the barber, who must shave himself.

Third, these problems ***are real problems***. These paradoxes go to the core of mathematics and logic. And since mathematics and logic are a part (an extremely important part) of the ***real world***, these problems are ***real***.

Yes, you mentioned the relationship to Cantor and as I said, so what? Also in the podcast they talked about various types of paradoxes, and yet they were all still paradoxical, weren't they? And if that doesn't reflect a variety of realistic pretensions, what does.Of course there is not actually a place where all are cleanly shaven, etc. But if there was no pretension that there could be, there'd be no point or purpose to the paradox. You don't seem to get that at all. You pontificate that mathematics and logic are part of the quote unquote real world, as if we don't already know we invented them to measure and understand it. So of course the problems we apply them to are real. Your whole explication here is trite.And worse, you didn't even understand the paradox that you took such trouble to to explain to Dave, writing as follows:"Now you want to say there is something wrong with this picture. What, exactly, is it? We defined the sets in an appropriate way and our reasoning is valid. Certainly you cannot simply deny that there is a problem because it is unseemly (and it IS unseemly). So, please demonstrate where I have gone wrong."And when I demonstrated it instead, you said nothing, and still haven't addressed the mistakes you made there.Massimo can cover for you and say it didn't represent the real world, etc., but then if it didn't (in spite of having said it did), what was it you were trying to validly explain? (And put in a lot of quote marks if that helps you.)

"Of course there is not actually a place where all are cleanly shaven, etc. But if there was no pretension that there could be, there'd be no point or purpose to the paradox. You don't seem to get that at all."

Yes, a pretense "that there could be" such a place must exist if there is to be a relevant paradox. But in the case of the barber paradox, that pretense is only pretense and nothing more, rendering the the "paradox" harmless. We can always negate the premise that leads to the contradiction -- the existence of the barber having the said properties. The barber paradox should only be seen as an elucidation of a more abstract problem. The place where the pretense becomes harmful is in naive set theory, or what set theorists believed around the time of Cantor. And in that place, the relevant paradox is Russell's paradox. Only in this case there is more than mere pretense -- Russell's paradox is inevitable, as long as you subscribe to naive set theory. And we correct the problem, that is, avoid the paradox, in much the same way: We take an axiom that says no set can contain itself. Only it's not quite so easy, because set theory needs to accommodate collections such as "the collection of all sets", which cannot be a set, else it would have to contain itself. Such collections are called proper classes, where a proper class is defined as any class (collection) that is not a set.

You seem to want to dismiss any reference to Cantor, but there was a real problem with naive set theory, so these references are not irrelevant. Why do you avoid this issue?

I'm not avoiding the issue with Cantor, just saying it was irrelevant to the efforts you made to explain the paradox to Dave. Because a paradox is a logical illusion, and when you explained it, you found ourself explaining the illusion instead of the logic. Which in effect destroyed the illusion. Which I don't think was your purpose.

That's completely incorrect. I responded to DaveK exactly twice, and never mentioned Cantor or Russell in those responses. Furthermore, you didn't reply to either of those responses. Possibly, you have confused me with Eamon.

I don't think I destroyed any illusion because there was no illusion to destroy. Russell's paradox is a real problem in naive set theory, but the real problem is destroyed by incorporating an appropriate limiting axiom. So right now, you are making about as much sense as DaveK.

Didn't mean to confuse the situation, just being brief. I did a post just now above to answer some of this. A paradox is just a lack of logic, and as physics is reasonably self-consistent as far as we know, it is logical to that extent. So a paradox is as much a matter pf physics as it is of logic. Science is comprised of physics and stamp collecting, and logic is just abstract waffle if it offends physics.

DaveK, you kept talking about "conflating" before, but you are conflating science and logic. Logic is a tool for partitioning statements (ideally in some formal language, but not always) into true, false, and indeterminate, based on assumptions. Science is the technique of finding theories that concur with observations. The assumptions in logic need not be empirical at all.

Paradox is not "lack of logic". Russell's paradox exists in the presence of logic. Another good example is the Burali-Forti paradox: The set of all ordinal numbers has all the properties of an ordinal number, and therefore must be an ordinal number, therefore it must have a successor which is an ordinal, therefore it must include that successor, therefore it must be greater than its own successor, therefore it must be greater than itself. It's perfectly logical from the premises of naive set theory, as is the Russell paradox.

What's missing in these cases is not logic. If you throw out logic, you have no paradox, because you cannot infer the paradoxical conclusions. What's missing is a sufficiently limiting axiom to prevent the paradox from occurring.

It is a matter of defining the terms in any way you choose, but you require those terms to be logical, whether it in physics, liar cases, set theory, or whatever. If they are paradoxical they are by definition illogical. The terms fail to follow an unbroken logical thread, and thus a 'paradox' arises, and so the terms need revision to straighten them out. You are just restating what I have said and somehow concluded that to break logic requires logic, thus a paradox is logical. If your paradox is logical, where is the paradox?

Reading your set theory example, and others here, and the equations, is is just another definitional conundrum. You can chase any rabbit into any maze, propose this and that logical conundrum based on the terms you imagine, and try to figure why those terms should produce that illogicality. Best of luck, but apply it to empirical results, preferably, that's all I'm saying, as that might straighten out the terms.

You are wrong and you are just being argumentative. Paradoxes arise from faulty axioms or premises, not from faulty logic. If the latter were the case, a paradox would be of no interest. As it is, when seemingly consistent axioms, through sound logic, yield contradictions, we are forced to reexamine the premises. And that *is* interesting.

The reference to empiricism is irrelevant. If we encounter contradiction in a physical experiment, and the result is repeatable, it is our assumptions about the physical world that are to be questioned, not the logic. That is why physicists no longer believe in the aether.

I know you have a lot invested in your point of view, but you should step back and reexamine, instead of continuing to dig deeper into that hole.

A final effort to see if I can bring Baron's correction to bear. Take a statement as a whole of various statements (axioms) that might in themselves have self-consistency or application to reality, but together they make a self-contradictory whole. Perhaps reality is warped, or the statement as a whole is simply self-contradictory (as in your abstract paradox cases).

You can replace or re-arrange the axioms (and any assumed realities on which they are based) to make it self-consistent, or you can dwell on how (un)special it is that they form a self-contradictory whole. Either reality is warped or your logic is warped, depending how abstract your analysis might be. I prefer the former. In fact, you literally dig a hole by shifting the issue to axioms that themselves should be logical or empirical. Like turtles, it's logic all the way up and down - just logic (or lack of it), and preferably applied to reality. I'm done with this head banging repetition.

Massimo, I know (I think) that you're a passionate advocator of bayesian inference. It would be very interesting to here you sum up how bayesian inference (or any other multi-valued logic for that matter) can be an alternative to paraconsistent logic. As I understand it, the bayesian solution has some advantages in terms of intuitivity. Am I right?Thanks in advance!

> Of course there is not actually a place where all are cleanly shaven, etc. But if there was no pretension that there could be, there'd be no point or purpose to the paradox. You don't seem to get that at all. <

Oh my. No, Baron, you keep not getting it. Let me try something different: mathematicians study all sorts of structures that do not exist in the physical world, for instance high dimensional manifolds. They present *real* problems of a *mathematical* nature, but they have nothing at all to do with physics or the actual world. Logic is the same: a lot of problems in logic are, well, logical, not physical or biological.

denlillekemisten,

yes, you are correct, I am sympathetic to Bayesian analysis. But I don't see how that would help with paradoxes. Bayesianism is an approach useful for probabilistically based decision making, not a logical tool.

Oh my no, Massimo, your mathematics example serves no purpose at all when it comes to understanding paradoxes, and this one in particular. The pretense that they ARE creatures of the physical world is the key.

You say a lot of problems in logic are not physical or biological. Clearly a non sequitur when we're talking about imaginary barbers don't you think?

In the alternatives section of the PL wiki page there is discussion of how these problems can be avoided without getting rid of the classical logic rules. I think this is a much more intuitive approach to the problem as well.

I have to say this podcast was very thought provoking. It gave me kind of the same experience that Massimo mentioned on the show: I have a full science training in physics but have never heard of Logics in the plural. So understandably I was quite confused a lot of the time. What bothered me most are the logical facts that were mentioned as things against which a particular logic can be checked. As Julia said in most other endeavours we actually have some "reality" against we can check our theory but I don't see how that works in logic.The grammar example mentioned works as follows in my mind: We come up with a grammar, construct some sentences from it and then show it to a native speaker who will tell us if they are intelligible and correct. (If that is not how it works please correct me.) My problem now is that I don't know what would be the equivalent of a native speaker in the case of logical facts. As Julia mentioned we can not just go out and ask people if statements such as "sth. can be true and false" are correct, because, well, intuition doesn't serve us very well as graham priest mentioned, as we are not used to such examples. What we then could do is do some reasoning to see if the sentence is correct. But to check if the reasoning is correct we presumably have to subscribe to some logic.In my mind this is a catch-22, because you end up checking a theory against facts that were determined on the basis of the theory. In that case you have checked if the theory is self consistent (or consistent with another logic), but I don't see how this can discriminate between theories.

As I said I don't know anything about logics, so any help in understanding this is appreciated

I am not all that knowledgeable either, but it seems that what we check a given logic against is what inferences it would permit about various kinds of facts (metaphysical, physical, mathematical...).

For example, take the logical statement "If A then B; A, therefore C." This can't be right because it would license inferences like "If I have a sister then she is female; I have a sister, therefore JFK was a manatee." So the set of all logics that contain "If A then B; A, therefore C" as a theorem must be wrong.

Thanks for your answer Ian. At first glance I thought I had understood it. :)

On second thought however I have some doubts. Such statements as the one you provided are similar to what I guess logical explosion can do. But since explosion seems to be up for debate the statement "this can't be right" does not necessarily follow for me (at least not in this discussion. If we were talking in any other context I would completely agree with you). I guess there is some subtle difference I am missing.

Again my problem is with the term "can't be right". I don't know how to assess that for a logical statement without using some form of logic.

I came up with some analogy to science which might be useful, but I am not sure if it holds. In science we establish facts with our current best measurement methods. We use these to test theories, which in turn might allow us to improve our measurements. Then it might turn out that some facts we used were wrong. So we build new theories on the basis of the new facts.In Logic that would probably mean we start from a logic build on our intuition and test it against common sense. As we progress with the development of our logic we hit paradoxes and need to improve our logics to accommodate these. When we have trained enough on such paradoxes we might end up with an improved common sense to test our logics against.

This feels weird to me, because logic always was kind of an absolute thing to me...But I don't see an obvious reason why it should be.

One place to start might be here: http://en.wikipedia.org/wiki/Three-valued_logicor here:http://en.wikipedia.org/wiki/Many-valued_logic

In particular, look at the sections on Kleene (K3) and Priest (P3) three-valued logics. We can determine by examination whether these logics meet the usual rules of inference. For example (and I'm getting out of my comfort zone so take this with a grain of salt) I think Priest's logic (the same logic discussed in the podcast) probably does not support deduction by means of reductio ad absurdum or modus tollens. Reductio ad absurdum fails because if A implies B, and A implies not B, we can't conclude that A is false, since B could be a dialetheia (both true and false). Modus tollens fails because if A implies B, and B is false, we can't conclude that A is false, because B could be a dialetheia and therefore true at the same time. As I said in an earlier post, paraconsistent logic would have to be weaker (able to prove less from a given set of premises) than classical logic, to avoid explosion, and not having these two rules to work with is a manifestation of that weakening.

I think that Kleene logic, on the other hand, supports most if not all of the classical rules of inference. It does require that law of excluded middle is not valid in the logic. The middle value of K3 logic has the semantics of "either true or false, but we don't know which".

I know this doesn't really answer your question, but it might help to know that, while not all logics are the same, there is a common basis in the rules of inference (or appropriate subsets thereof) which makes them useful.

Then in response I can only say that I am not being argumentative, and by claiming I am, you are being argumentative. But that does not advance the discussion. Nor does the rest of the post because you did not read Baron's correction of your attempt to define as lack of logic as having logic, nor my correction of it. Empriicism is entriely relevant because if an axiom does not apply to reality, what use is it other than as as self-consistent statement?

The 'hole' you hope exists simply does not, and no amount of rhetoric on your part can change that. The abiding rule is self-consistentcy, read any definition of 'paradox' and it will offend self-consistentcy and therefore it is not logical. It's a pity you opt for the rhetoric of Eamon and Massimo, so I will bow out of this pointless argument. It's better not to inflate, conflate, uselessly redefine, rely on contradicitory statements, and raise nonsense to being special (such as paradoxes and emergence).

PS You aren't even reading the thread correctly. Baron never responded to me, except when he mistook me for Eamon, and that reply was not at all along the lines of what you just claimed above. At any rate he was presenting a rhetorical argument (like yours) and not a logical one, so it hardly changes anything. (I'm being generous with "hardly").

Wrong again Richard, and every occasion I have responded to you, to the extent I have responded (I like to avoid most of the abstracts). This is the Baron quote, and you will need to read your own explanations to me referred to by Baron to understand it. You have apprently again completely misunderstood the reference and concluded I must have no relevant quote by Baron to refer to. It is serious overstepping once again, and similar to massino, eamon and others who cannot stay with the argument.

"I'm not avoiding the issue with Cantor, just saying it was irrelevant to the efforts you made to explain the paradox to Dave. Because a paradox is a logical illusion, and when you explained it, you found ourself explaining the illusion instead of the logic. Which in effect destroyed the illusion. Which I don't think was your purpose."

Im not going to revisit your explanations, Baron's problem with them, my problem with them, Cantor, or the rest. That's why this is all pointless, it reduces to that. Just remember, you can't pass the logical buck onto this axiom or that, and their failings at making a self-consistent whole. If they fail, they fail, like turtles, its logic all the way down.

"During the early Middle Ages, virtually all scholars maintained the spherical viewpoint first expressed by the Ancient Greeks. By the 14th century, belief in a flat earth among the educated was nearly nonexistent."